Newtonian potential theory for unbounded sources and applications to free boundary problems
type: Item
instance of: scholarly article
title: Newtonian potential theory for unbounded sources and applications to free boundary problems
zbMATH Open document ID: 0876.35136
DOI: 10.1007/BF02820440
author: Lavi Karp
author: Avmir S. Margulis
published in: Journal d'Analyse Mathématique
publication date: 1997-12-03T00:00:00Z
review text: The authors study comprehensively the most important inverse problems in the Newtonian potential theory in \(\mathbb{R}^n\). They formulate this problem in the following terms: Let \(D\) be an open set in \(n\)-dimensional Euclidean space \(\mathbb{R}^n\), \(n\geq 2\), and let \(\sigma\) be a positive, bounded function in \(\mathbb{R}^n\). Given a harmonic function \(v\) in \(D\), find a function \(f\) and an open set \(\Omega\) satisfying the system: \[ \Delta f(x)=-\sigma(x) \chi_\Omega(x),\quad x\in\mathbb{R}^n,\quad f(x)= v(x),\quad x\in D\setminus\Omega; \] \(\Delta\) is the Laplace operator in the distributional sense and \(\chi_\Omega\) is the characteristic function of \(\Omega\). The main object of the paper is the regularity of the free boundary \(\Gamma=\partial\Omega\). The authors present results of other authors, their own ones from the last 15 years, compare and comment them and show their interconnections. The paper contains five sections, where the authors present their most important results.
Mathematics Subject Classification ID: 35R35
Mathematics Subject Classification ID: 31-02
Mathematics Subject Classification ID: 31B20
zbMATH DE Number: 1019374
zbMATH Keywords: inverse problems in the Newtonian potential theory
zbMATH Keywords: generalized Newtonian potential
reviewed by: Vasile Ionescu
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02820440
OpenAlex ID: W1971291053
description: scientific article; zbMATH DE number 1019374